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characteristic or principal. And thus, whereas he made his de-
ductions by setting out with the two equations of a ray, I on the
other hand establish and employ the one equation of a system.
'The function which I have introduced for this purpose, and
made the basis of my method of deduction in mathematical
Optics, had, in another connexion, presented itself to former
writers as expressing the result of a very high and extensive
induction in that science. This known result is usually called the
law of least action, but sometimes also the principle of least time
[see chapter on Fermat], and includes all that has hitherto been
discovered respecting the rules which determine the forms and
positions of the lines along which light is propagated, and the
changes of direction of those lines produced by reflection or
refraction, ordinary or extraordinary [the latter as in a doubly
refracting crystal, say Iceland spar, in which a single ray is
split into two, both refracted, on entering the crystal]. A
certain quantity which in one physical theory is the action, and
in another the tim e, expended by light in going from any first
to any second point, is found to be less than if the light had
gone in any other than its actual path, or at least to have what
is technically called its variation null, the extremities of the
path being unvaried. The mathematical novelty of my method
consists in considering this quantity as a function of the co-
ordinates of these extremities, which varies when they vary,
according to a law which I have called the law of varying action;
and in reducing all researches respecting optical systems of rays to
the study of this single function: a reduction which presents
mathematical Optics under an entirely novel view, and one
analogous (as it appears to me) to the aspect under which
Descartes presented the application of Algebra to Geometry.9
Nothing need be added to this account of Hamilton's, except
possibly the remark that no science, no matter how ably ex-
pounded, is understood as readily as any novel, no matter how
badly written. The whole extract will repay a second reading.
In this great work on systems of rays Hamilton had builded
better than even he knew. Almost exactly 100 years after the
above abstract was written the methods which Hamilton intro-
duced into optics were found to be just what was required hi